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\author{Class 2019 Math and Applied Math }
\title{Applied stochastic processes - Homework 01}
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%\date{2021 年 2 月 28 日}
\date{March 9, 2021}

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%\subsection{Homework 01}
%E3.1.2, P3.1.4, E3.2.2, P3.2.4, E3.3.2, P3.3.6.

\begin{document}

\maketitle

\begin{enumerate}
\item [E3.1.2.]  A Markov chain $X_0,X_1,X_2,\cdots$ has the following transition probability matrix 
\begin{eqnarray*}
P=
\begin{blockarray}{cccc}
& 0 & 1 & 2 \\
\begin{block}{c[ccc]}
  0 & 0.7 & 0.2 & 0.1 \\
  1 & 0    & 0.6 & 0.4 \\ 
  2 & 0.5 & 0   &  0.5 \\
\end{block}
\end{blockarray}.
\end{eqnarray*}
Determine the conditional probabilities $$\mathbb{P}\{X_2 =1,X_3 =1\mid X_1 =0\} \text{ and } \mathbb{P}\{X_1 =1,X_2 =1\mid X_0 =0\}.$$


\item  [P3.1.4.] The random variables $\xi_1,\xi_2,\cdots$ are independent and with the common probability mass function
\begin{table}[ht]
\centering
\begin{tabular}{c|cccc} %\hline
$k$ & 0 & 1 & 2 & 3 \\ \hline 
$\mathbb{P}\{\xi = k\}$ & 0.1 & 0.3 & 0.2 & 0.4 \\  %\hline
\end{tabular}
\end{table}
Set $X_0 = 0$, and let $X_n = \max\{\xi_1,\cdots, \xi_n\}$ be the largest $\xi$ observed to date. Determine the transition probability matrix for the Markov chain $\{X_n\}$.


\item [E3.2.2.] A particle moves among the states $0, 1, 2$ according to a Markov process whose
transition probability matrix is
\begin{eqnarray*}
P=
\begin{blockarray}{cccc}
& 0 & 1 & 2 \\
\begin{block}{c[ccc]}
  0 & 0 & 0.5 & 0.5 \\
  1 & 0.5  & 0 & 0.5 \\ 
  2 & 0.5 & 0.5  & 0 \\
\end{block}
\end{blockarray}.
\end{eqnarray*}
Let $X_n$ denote the position of the particle at the $n$th move. Calculate $\mathbb{P}\{X_n =0\mid X_0 = 0\}$ for $n = 0,1,2,3,4$.


\item  [P3.2.4.] Suppose $X_n$ is a two-state Markov chain whose transition probability matrix is 
\begin{eqnarray*}
P=
\begin{blockarray}{ccc}
& 0 & 1  \\
\begin{block}{c[cc]}
  0 & \alpha & 1-\alpha  \\
  1 & 1-\beta  & \beta \\ 
\end{block}
\end{blockarray}.
\end{eqnarray*}
Then, $Z_n = (X_{n-1},X_n)$ is a Markov chain having the four states $(0,0)$, $(0,1)$, $(1,0)$, and $(1,1)$. 
Determine the transition probability matrix.


\item [E3.3.2.] Consider two urns A and B containing a total of $N$ balls. An experiment is performed in which a ball is selected at random (all selections equally likely) at time $t$ ($t = 1, 2, \cdots $) from among the totality of $N$ balls. Then, an urn is selected at random (A is chosen with probability $p$ and B is chosen with probability $q$) and the ball previously drawn is placed in this urn. The state of the system at each trial is represented by the number of balls in A. Determine the transition matrix for this Markov chain.


\item [P3.3.6.] Two teams, A and B, are to play a best of seven series of games. Suppose that the outcomes of successive games are independent, and each is won by A with probability $p$ and won by B with probability $1 - p$. Let the state of the system be represented by the pair $(a,b)$, where $a$ is the number of games won by A, and $b$ is the number of games won by B. Specify the transition probability matrix. Note that $a+b \le 7$ and that the entries end whenever $a = 4$ or $b = 4$.

\end{enumerate}


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\end{document}

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\subsection{Homework 01}
E3.1.2, P3.1.4, E3.2.2, P3.2.4, E3.3.2, P3.3.6.

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\subsection{Homework 02}
E.3.4.1, E3.4.2, P3.4.1, P3.4.5, E3.5.1, P3.5.1. 

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\subsection{Homework 03}
E4.1.10, P4.1.1, P4.1.5, E4.3.1, E4.3.2, E4.4.2.

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\subsection{Homework 04}
E5.1.1, E5.1.7, P5.1.10, E5.2.1, P5.2.1.

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\subsection{Homework 05}
E5.3.1, E5.3.3, E5.3.7, P5.3.1, E5.4.1, E5.4.3. 

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\subsection{Homework 06}
E6.1.1, E6.1.2, P6.1.1, P6.1.2.

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\subsection{Homework 07}
E7.1.2, E7.1.3, E7.2.1, E7.2.3, P7.2.1.

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\subsection{Homework 08}
E8.1.1, E8.1.2, E8.1.4, P8.1.1, P8.1.3, E8.2.1.

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